Question

(a) State whether or not the equation is autonomous. (b) Identify all equilibrium solutions (if any). (c) Sketch the direction field for the differential equation in the rectangular portion of the \(t y\)-plane defined by \(-2 \leq t \leq 2,-2 \leq y \leq 2\). $$ y^{\prime}=-1 $$

Short answer

Yes, the given differential equation is autonomous since it can be written in the form \(y' = f(y)\), where the function \(f\) does not depend on the independent variable \(t\). b) What are the equilibrium solutions for the given differential equation? There are no equilibrium solutions for the given differential equation since the derivative is never zero. c) How does the direction field look like for the rectangular portion of the \(ty\)-plane defined by \(-2 \leq t \leq 2, -2 \leq y \leq 2\)? The direction field displays a uniform slope of -\(1\) over the specified rectangular portion of the \(ty\)-plane.

Step-by-step solution

Determine if the equation is autonomous

A differential equation is autonomous if it can be written in the form \(y' = f(y)\), where the function \(f\) does not depend on the independent variable \(t\). In this case, the given equation is: $$ y' = -1 $$ Since the right-hand side of the equation does not involve \(t\), the given equation is autonomous.

Identify equilibrium solutions

An equilibrium solution is a constant function \(y(t)=c\) that is a solution to the differential equation. In this case, if \(y(t)=c\) is constant, its derivative \(y'(t)=0\). However, the given equation states that \(y'=-1\). This means there are no equilibrium solutions for the given equation since the derivative is never zero.

Sketch the direction field

The direction field is a graphical representation of the given differential equation that shows the slope of the function at each point in the \(ty\)-plane. To sketch the direction field for the rectangular portion of the \(ty\)-plane defined by \(-2 \leq t \leq 2, -2 \leq y \leq 2\), we can note that the slopes are always the same, which is -1, regardless of the value of \(t\) and \(y\). This means that there will be a uniform direction field over the specified rectangular portion of the \(ty\)-plane, with the slope of -\(1\) at every point. So, draw horizontal lines at integer values of \(y\) from \(-2\) to \(2\), then on each of them draw short segments at integer values of \(t\) from \(-2\) to \(2\) with a slope of -\(1\). The direction field should look as follows: ``` | \ | \ | \ | | \ | \ | \ | ---------+--------- | \ | \ | \ | | \ | \ | \ | ---------+--------- | \ | \ | \ | | \ | \ | \ | ---------+--------- | \ | \ | \ | | \ | \ | \ | ``` Thus, the direction field would display a uniform slope of -\(1\) over the specified rectangular portion of the \(ty\)-plane.

Key concepts

Autonomous Equations

An autonomous differential equation is a special type of equation where the rate of change of a function is entirely dependent on the function itself and not on the independent variable. This means that if you have a differential equation of the form \( y' = f(y) \), it is autonomous because the rate of change \( y' \) depends solely on \( y \) and not on any other factors like time \( t \).
In practical terms, autonomy implies that the system's behavior is consistent regardless of the point in time or space where it begins. This consistency allows autonomous equations to often model real-life phenomena where external factors don't influence the rate of change, such as population growth in an unchanging environment.

• Example: The equation \( y' = -1 \) is autonomous because the equation indicates that the rate of change is a constant \(-1\), unaffected by \(t\). This consistency characterizes autonomy in differential equations.
• When solving problems involving autonomous equations, students should always verify the presence of the independent variable to conclude if the equation is autonomous.

Equilibrium Solutions

Equilibrium solutions in the context of differential equations are constant solutions, where the rate of change \( y' \) equals zero. That means the function \( y(t) = c \) does not change over time; it remains constant at \( c \).
Why It's Important: Finding equilibrium solutions helps in understanding stability within systems. If a system settles at an equilibrium state, it is stable. If it moves away from equilibrium, it is unstable.
However, not all differential equations have equilibrium solutions. For instance, with the autonomous equation \( y' = -1 \), the rate of change is always \(-1\), which means no constant solution could satisfy this equation since it cannot achieve a rate of change of zero.

• For equilibrium solutions, set \( y' = 0 \) and solve for \( y \).
• If there are no possible solutions, the system does not have equilibrium solutions.

Recognizing and understanding equilibrium can illuminate how the system reacts to changes over time.

Direction Fields

Direction fields, also known as slope fields, provide a visual method to understand differential equations by plotting slopes at various points in the \( ty \)-plane. They help indicate the behavior of solutions without solving the equation analytically.
In the context of the problem, the direction field must be sketched for \(-2 \leq t \leq 2\) and \(-2 \leq y \leq 2\) for the equation \( y' = -1 \). This equation's uniform slope of \(-1\) across all points suggests that at every position in the rectangle, the solution's graph tends to fall at an angle representing that slope.
How to Create a Direction Field:

• Choose several points in the \( ty \)-plane.
• At each point, draw a short line segment perpendicular to the \( y \)-axis with a slope equal to the derivative \( y' \).
• In the case of \( y' = -1 \), every segment should be parallel at the same negative angle.

Drawing these tiny segments across the plane gives a comprehensive idea of how solutions move forward from any starting point, thus providing a powerful visualization tool.